Interacting elliptic inclusions by the method of complex potentials

An accurate series solution has been obtained for a piece-homogeneous elastic plane containing a finite array of nonoverlapping elliptic inclusions of arbitrary size, aspect ratio, location and elastic properties. The method combines standard Muskhelishvili s representation of general solution in terms of complex potentials with the superposition principle and newly derived re-expansion formulae to obtain a complete solution of the many-inclusion problem. By exact satisfaction of all the interface conditions, a primary boundary-value problem stated on a complicated heterogeneous domain has been reduced to an ordinary well-posed set of linear algebraic equations. A properly chosen form of potentials provides a remarkably simple form of solution and thus an efficient computational algorithm. The theory developed is rather general and can be applied to solve a variety of composite mechanics problems. The advanced models of composite involving up to several hundred inclusions and providing an accurate account for the microstructure statistics and fiber–fiber interactions can be considered in this way. The numerical examples are given showing high accuracy and numerical efficiency of the method developed and disclosing the way and extent to which the selected structural parameters influence the stress concentration at the matrix–inclusion interface.